Quantum operations and channels

Quantum operations and channels covers the physically allowed transformations of quantum states, such as isometries and completely positive maps, and impossibility results like the no-cloning theorem.

TheoremNo-cloning theorem for two states#✓fully proved

Let \(H\) be a complex inner product space and let \(\psi, \varphi, b \in H\) be unit vectors with \(0 < |\langle \psi, \varphi \rangle| < 1\). Then there is no \(\mathbb{C}\)-linear isometric isomorphism \(U\) of \(H \otimes_{\mathbb{C}} H\) onto itself with \(U(\psi \otimes b) = \psi \otimes \psi\) and \(U(\varphi \otimes b) = \varphi \otimes \varphi\).

public theorem no_cloning_two_state
    {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
    (ψ φ b : H) (_hψ : ‖ψ‖ = 1) (_hφ : ‖φ‖ = 1) (hb : ‖b‖ = 1)
    (hpos : 0 < ‖⟪ψ, φ⟫_ℂ‖) (hlt : ‖⟪ψ, φ⟫_ℂ‖ < 1) :
    ¬ ∃ (U : (H ⊗[ℂ] H) ≃ₗᵢ[ℂ] (H ⊗[ℂ] H)),
      U (ψ ⊗ₜ[ℂ] b) = ψ ⊗ₜ[ℂ] ψ ∧ U (φ ⊗ₜ[ℂ] b) = φ ⊗ₜ[ℂ] φ

Sources

• W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned", Nature 299 (1982), 802-803, DOI 10.1038/299802a0.

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